arXiv subjects

JC Olivier

Publications and source records attributed to JC Olivier.

4 recordsLinked to original sources

arXiv2022

On minimum phase transformation and filter design

Minimum-phase finite impulse response filters are widely used in practice, and much research has been devoted to the design of such filters. However, for the important case of Chebyshev filters there is a curious mismatch between current best practice and well-established theoretical principles. The paper shows that this difference can be understood through analysis of the time-domain factorization of a suitable extended matrix. This analysis explains why the definition of a factorable linear phase filter must be revised. The time domain analysis of factorization suggests initial values leading to fast and accurate convergence of iterative algorithms for the design of minimum-phase finite impulse response filters. Numerical results are provided to demonstrate that a significant improvement in filter tap accuracy is obtained when the well-established theoretical principles are correctly applied to the design of minimum phase finite impulse response filters.

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arXiv2020

Optimizing short stabilizer codes for asymmetric channels

For a number of quantum channels of interest, phase-flip errors occur far more frequently than bit-flip errors. When transmitting across these asymmetric channels, the decoding error rate can be reduced by tailoring the code used to the channel. However, analyzing the performance of stabilizer codes on these channels is made difficult by the #P-completeness of optimal decoding. To address this, at least for short codes, we demonstrate that the decoding error rate can be approximated by considering only a fraction of the possible errors caused by the channel. Using this approximate error rate calculation, we extend a recent result to show that there are a number of $[[5\leq n\leq12,1\leq k\leq3]]$ cyclic stabilizer codes that perform well on two different asymmetric channels. We also demonstrate that an indication of a stabilizer code's error rate is given by considering the error rate of a classical binary code related to the stabilizer. This classical error rate is far less complex to calculate, and we use it as the basis for a hill climbing algorithm, which we show to be effective at optimizing codes for asymmetric channels. Furthermore, we demonstrate that simple modifications can be made to our hill climbing algorithm to search for codes with desired structure requirements.

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arXiv2019

Heuristic construction of codeword stabilized codes

The family of codeword stabilized codes encompasses the stabilizer codes as well as many of the best known nonadditive codes. However, constructing optimal $n$-qubit codeword stabilized codes is made difficult by two main factors. The first of these is the exponential growth with $n$ of the number of graphs on which a code can be based. The second is the NP-hardness of the maximum clique search required to construct a code from a given graph. We address the second of these issues through the use of a heuristic clique finding algorithm. This approach has allowed us to find $((9,97\leq K\leq100,2))$ and $((11,387\leq K\leq416,2))$ codes, which are larger than any previously known codes. To address the exponential growth of the search space, we demonstrate that graphs that give large codes typically yield clique graphs with a large number of nodes. The number of such nodes can be determined relatively efficiently, and we demonstrate that $n$-node graphs yielding large clique graphs can be found using a genetic algorithm. This algorithm uses a novel spectral bisection based crossover operation that we demonstrate to be superior to more standard crossover operations. Using this genetic algorithm approach, we have found $((13,18,4))$ and $((13,20,4))$ codes that are larger than any previously known code. We also consider codes for the amplitude damping channel. We demonstrate that for $n\leq9$, optimal codeword stabilized codes correcting a single amplitude damping error can be found by considering standard form codes that detect one of only three of the $3^{n}$ possible equivalent error sets. By combining this error set selection with the genetic algorithm approach, we have found $((11,68))$ and $((11,80))$ codes capable of correcting a single amplitude damping error and $((11,4))$, $((12,4))$, $((13,8))$, and $((14,16))$ codes capable of correcting two amplitude damping

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arXiv2019

Modified belief propagation decoders for quantum low-density parity-check codes

Quantum low-density parity-check codes can be decoded using a syndrome based $\mathrm{GF}(4)$ belief propagation decoder. However, the performance of this decoder is limited both by unavoidable $4$-cycles in the code's factor graph and the degenerate nature of quantum errors. For the subclass of CSS codes, the number of $4$-cycles can be reduced by breaking an error into an $X$ and $Z$ component and decoding each with an individual $\mathrm{GF}(2)$ based decoder. However, this comes at the expense of ignoring potential correlations between these two error components. We present a number of modified belief propagation decoders that address these issues. We propose a $\mathrm{GF}(2)$ based decoder for CSS codes that reintroduces error correlations by reattempting decoding with adjusted error probabilities. We also propose the use of an augmented decoder, which has previously been suggested for classical binary low-density parity-check codes. This decoder iteratively reattempts decoding on factor graphs that have a subset of their check nodes duplicated. The augmented decoder can be based on a $\mathrm{GF}(4)$ decoder for any code, a $\mathrm{GF}(2)$ decoder for CSS code, or even a supernode decoder for a dual-containing CSS code. For CSS codes, we further propose a $\mathrm{GF}(2)$ based decoder that combines the augmented decoder with error probability adjustment. We demonstrate the performance of these new decoders on a range of different codes, showing that they perform favorably compared to other decoders presented in literature.

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